Level of measurement
Level of measurement or scale of measure is a classification that describes the nature of information within the values assigned to variables. Psychologist Stanley Smith Stevens developed the best-known classification with four levels, or scales, of measurement: nominal, ordinal, interval, and ratio. This framework of distinguishing levels of measurement originated in psychology and is widely criticized by scholars in other disciplines. Other classifications include those by Mosteller and Tukey, and by Chrisman.
Stevens proposed his typology in a 1946 Science article titled "On the theory of scales of measurement". In that article, Stevens claimed that all measurement in science was conducted using four different types of scales that he called "nominal", "ordinal", "interval", and "ratio", unifying both "qualitative" (which are described by his "nominal" type) and "quantitative" (to a different degree, all the rest of his scales). The concept of scale types later received the mathematical rigour that it lacked at its inception with the work of mathematical psychologists Theodore Alper (1985, 1987), Louis Narens (1981a, b), and R. Duncan Luce (1986, 1987, 2001). As Luce (1997, p. 395) wrote:
S. S. Stevens (1946, 1951, 1975) claimed that what counted was having an interval or ratio scale. Subsequent research has given meaning to this assertion, but given his attempts to invoke scale type ideas it is doubtful if he understood it himself ... no measurement theorist I know accepts Stevens's broad definition of measurement ... in our view, the only sensible meaning for 'rule' is empirically testable laws about the attribute.
|Nominal||Classification, membership||=, ≠||Grouping||Mode||Qualitative variation|
|Ordinal||Comparison, level||>, <||Sorting||Median||Range,|
|Interval||Difference, affinity||+, −||Comparison to a standard||Arithmetic mean||Deviation|
|Ratio||Magnitude, amount||×, /||Ratio||Geometric mean,||Coefficient of variation,|
The nominal type differentiates between items or subjects based only on their names or (meta-)categories and other qualitative classifications they belong to; thus dichotomous data involves the construction of classifications as well as the classification of items. Discovery of an exception to a classification can be viewed as progress. Numbers may be used to represent the variables but the numbers do not have numerical value or relationship: for example, a globally unique identifier.
Examples of these classifications include gender, nationality, ethnicity, language, genre, style, biological species, and form. In a university one could also use hall of affiliation as an example. Other concrete examples are
- in grammar, the parts of speech: noun, verb, preposition, article, pronoun, etc.
- in politics, power projection: hard power, soft power, etc.
- in biology, the taxonomic ranks below domains: Archaea, Bacteria, and Eukarya
- in software engineering, type of faults: specification faults, design faults, and code faults
Nominal scales were often called qualitative scales, and measurements made on qualitative scales were called qualitative data. However, the rise of qualitative research has made this usage confusing. If numbers are assigned as labels in nominal measurement, they have no specific numerical value or meaning. No form of arithmetic computation (+, −, ×, etc.) may be performed on nominal measures. The nominal level is the lowest measurement level used from a statistical point of view.
Equality and other operations that can be defined in terms of equality, such as inequality and set membership, are the only non-trivial operations that generically apply to objects of the nominal type.
The mode, i.e. the most common item, is allowed as the measure of central tendency for the nominal type. On the other hand, the median, i.e. the middle-ranked item, makes no sense for the nominal type of data since ranking is meaningless for the nominal type.
The ordinal type allows for rank order (1st, 2nd, 3rd, etc.) by which data can be sorted but still does not allow for a relative degree of difference between them. Examples include, on one hand, dichotomous data with dichotomous (or dichotomized) values such as 'sick' vs. 'healthy' when measuring health, 'guilty' vs. 'not-guilty' when making judgments in courts, 'wrong/false' vs. 'right/true' when measuring truth value, and, on the other hand, non-dichotomous data consisting of a spectrum of values, such as 'completely agree', 'mostly agree', 'mostly disagree', 'completely disagree' when measuring opinion.
The ordinal scale places events in order, but there is no attempt to make the intervals of the scale equal in terms of some rule. Rank orders represent ordinal scales and are frequently used in research relating to qualitative phenomena. A student's rank in his graduation class involves the use of an ordinal scale. One has to be very careful in making a statement about scores based on ordinal scales. For instance, if Devi's position in his class is 10 and Ganga's position is 40, it cannot be said that Devi's position is four times as good as that of Ganga. The statement would make no sense at all. Ordinal scales only permit the ranking of items from highest to lowest. Ordinal measures have no absolute values, and the real differences between adjacent ranks may not be equal. All that can be said is that one person is higher or lower on the scale than another, but more precise comparisons cannot be made. Thus, the use of an ordinal scale implies a statement of 'greater than' or 'less than' (an equality statement is also acceptable) without our being able to state how much greater or less. The real difference between ranks 1 and 2, for instance, may be more or less than the difference between ranks 5 and 6. Since the numbers of this scale have only a rank meaning, the appropriate measure of central tendency is the median. A percentile or quartile measure is used for measuring dispersion. Correlations are restricted to various rank order methods. Measures of statistical significance are restricted to the non-parametric methods (R. M. Kothari, 2004).
In 1946, Stevens observed that psychological measurement, such as measurement of opinions, usually operates on ordinal scales; thus means and standard deviations have no validity, but they can be used to get ideas for how to improve operationalization of variables used in questionnaires. Most psychological data collected by psychometric instruments and tests, measuring cognitive and other abilities, are ordinal, although some theoreticians have argued they can be treated as interval or ratio scales. However, there is little prima facie evidence to suggest that such attributes are anything more than ordinal (Cliff, 1996; Cliff & Keats, 2003; Michell, 2008). In particular, IQ scores reflect an ordinal scale, in which all scores are meaningful for comparison only. There is no absolute zero, and a 10-point difference may carry different meanings at different points of the scale.
The interval type allows for the degree of difference between items, but not the ratio between them. Examples include temperature scales with the Celsius scale, which has two defined points (the freezing and boiling point of water at specific conditions) and then separated into 100 intervals, date when measured from an arbitrary epoch (such as AD), location in Cartesian coordinates, and direction measured in degrees from true or magnetic north. Ratios are not meaningful since 20 °C cannot be said to be "twice as hot" as 10 °C (unlike temperature in Kelvins), nor can multiplication/division be carried out between any two dates directly. However, ratios of differences can be expressed; for example, one difference can be twice another. Interval type variables are sometimes also called "scaled variables", but the formal mathematical term is an affine space (in this case an affine line).
Central tendency and statistical dispersion
The mode, median, and arithmetic mean are allowed to measure central tendency of interval variables, while measures of statistical dispersion include range and standard deviation. Since one can only divide by differences, one cannot define measures that require some ratios, such as the coefficient of variation. More subtly, while one can define moments about the origin, only central moments are meaningful, since the choice of origin is arbitrary. One can define standardized moments, since ratios of differences are meaningful, but one cannot define the coefficient of variation, since the mean is a moment about the origin, unlike the standard deviation, which is (the square root of) a central moment.
- See also: Positive real numbers § Ratio scale
The ratio type takes its name from the fact that measurement is the estimation of the ratio between a magnitude of a continuous quantity and a unit of measurement of the same kind (Michell, 1997, 1999). Most measurement in the physical sciences and engineering is done on ratio scales. Examples include mass, length, duration, plane angle, energy and electric charge. In contrast to interval scales, ratios can be compared using division. Very informally, many ratio scales can be described as specifying "how much" of something (i.e. an amount or magnitude). Ratio scale is often used to express an order of magnitude such as for temperature in Orders of magnitude (temperature).
Central tendency and statistical dispersion
The geometric mean and the harmonic mean are allowed to measure the central tendency, in addition to the mode, median, and arithmetic mean. The studentized range and the coefficient of variation are allowed to measure statistical dispersion. All statistical measures are allowed because all necessary mathematical operations are defined for the ratio scale.
Debate on Stevens's typology
While Stevens's typology is widely adopted, it is still being challenged by other theoreticians, particularly in the cases of the nominal and ordinal types (Michell, 1986). Some however have argued that the degree of discord can be overstated. Hand says, "Basic psychology texts often begin with Stevens's framework and the ideas are ubiquitous. Indeed, the essential soundness of his hierarchy has been established for representational measurement by mathematicians, determining the invariance properties of mappings from empirical systems to real number continua. Certainly the ideas have been revised, extended, and elaborated, but the remarkable thing is his insight given the relatively limited formal apparatus available to him and how many decades have passed since he coined them."
Duncan (1986) objected to the use of the word measurement in relation to the nominal type, but Stevens (1975) said of his own definition of measurement that "the assignment can be any consistent rule. The only rule not allowed would be random assignment, for randomness amounts in effect to a nonrule".
The use of the mean as a measure of the central tendency for the ordinal type is still debatable among those who accept Stevens's typology. Many behavioural scientists use the mean for ordinal data, anyway. This is often justified on the basis that the ordinal type in behavioural science is in fact somewhere between the true ordinal and interval types; although the interval difference between two ordinal ranks is not constant, it is often of the same order of magnitude.
For example, applications of measurement models in educational contexts often indicate that total scores have a fairly linear relationship with measurements across the range of an assessment. Thus, some argue that so long as the unknown interval difference between ordinal scale ranks is not too variable, interval scale statistics such as means can meaningfully be used on ordinal scale variables. Statistical analysis software such as SPSS requires the user to select the appropriate measurement class for each variable. This ensures that subsequent user errors cannot inadvertently perform meaningless analyses (for example correlation analysis with a variable on a nominal level).
L. L. Thurstone made progress toward developing a justification for obtaining the interval type, based on the law of comparative judgment. A common application of the law is the analytic hierarchy process. Further progress was made by Georg Rasch (1960), who developed the probabilistic Rasch model that provides a theoretical basis and justification for obtaining interval-level measurements from counts of observations such as total scores on assessments.
Other proposed typologies
Typologies aside from Stevens's typology have been proposed. For instance, Mosteller and Tukey (1977), Nelder (1990) described continuous counts, continuous ratios, count ratios, and categorical modes of data. See also Chrisman (1998), van den Berg (1991).
Mosteller and Tukey's typology (1977)
Mosteller and Tukey noted that the four levels are not exhaustive and proposed:
- Grades (ordered labels like beginner, intermediate, advanced)
- Ranks (orders with 1 being the smallest or largest, 2 the next smallest or largest, and so on)
- Counted fractions (bound by 0 and 1)
- Counts (non-negative integers)
- Amounts (non-negative real numbers)
- Balances (any real number)
For example, percentages (a variation on fractions in the Mosteller–Tukey framework) do not fit well into Stevens's framework: No transformation is fully admissible.
Chrisman's typology (1998)
Nicholas R. Chrisman introduced an expanded list of levels of measurement to account for various measurements that do not necessarily fit with the traditional notions of levels of measurement. Measurements bound to a range and repeating (like degrees in a circle, clock time, etc.), graded membership categories, and other types of measurement do not fit to Stevens's original work, leading to the introduction of six new levels of measurement, for a total of ten:
- Gradation of membership
- Extensive ratio
- Cyclical ratio
- Derived ratio
While some claim that the extended levels of measurement are rarely used outside of academic geography, graded membership is central to fuzzy set theory, while absolute measurements include probabilities and the plausibility and ignorance in Dempster–Shafer theory. Cyclical ratio measurements include angles and times. Counts appear to be ratio measurements, but the scale is not arbitrary and fractional counts are commonly meaningless. Log-interval measurements are commonly displayed in stock market graphics. All these types of measurements are commonly used outside academic geography, and do not fit well to Stevens' original work.
Scale types and Stevens's "operational theory of measurement"
The theory of scale types is the intellectual handmaiden to Stevens's "operational theory of measurement", which was to become definitive within psychology and the behavioral sciences, despite Michell's characterization as its being quite at odds with measurement in the natural sciences (Michell, 1999). Essentially, the operational theory of measurement was a reaction to the conclusions of a committee established in 1932 by the British Association for the Advancement of Science to investigate the possibility of genuine scientific measurement in the psychological and behavioral sciences. This committee, which became known as the Ferguson committee, published a Final Report (Ferguson, et al., 1940, p. 245) in which Stevens's sone scale (Stevens & Davis, 1938) was an object of criticism:
…any law purporting to express a quantitative relation between sensation intensity and stimulus intensity is not merely false but is in fact meaningless unless and until a meaning can be given to the concept of addition as applied to sensation.
That is, if Stevens's sone scale genuinely measured the intensity of auditory sensations, then evidence for such sensations as being quantitative attributes needed to be produced. The evidence needed was the presence of additive structure – a concept comprehensively treated by the German mathematician Otto Hölder (Hölder, 1901). Given that the physicist and measurement theorist Norman Robert Campbell dominated the Ferguson committee's deliberations, the committee concluded that measurement in the social sciences was impossible due to the lack of concatenation operations. This conclusion was later rendered false by the discovery of the theory of conjoint measurement by Debreu (1960) and independently by Luce & Tukey (1964). However, Stevens's reaction was not to conduct experiments to test for the presence of additive structure in sensations, but instead to render the conclusions of the Ferguson committee null and void by proposing a new theory of measurement:
Paraphrasing N. R. Campbell (Final Report, p.340), we may say that measurement, in the broadest sense, is defined as the assignment of numerals to objects and events according to rules (Stevens, 1946, p.677).
Stevens was greatly influenced by the ideas of another Harvard academic, the Nobel laureate physicist Percy Bridgman (1927), whose doctrine of operationism Stevens used to define measurement. In Stevens's definition, for example, it is the use of a tape measure that defines length (the object of measurement) as being measurable (and so by implication quantitative). Critics of operationism object that it confuses the relations between two objects or events for properties of one of those of objects or events (Hardcastle, 1995; Michell, 1999; Moyer, 1981a,b; Rogers, 1989).
The Canadian measurement theorist William Rozeboom (1966) was an early and trenchant critic of Stevens's theory of scale types.
Same variable may be different scale type depending on context
Another issue is that the same variable may be a different scale type depending on how it is measured and on the goals of the analysis. For example, hair color is usually thought of as a nominal variable, since it has no apparent ordering. However, it is possible to order colors (including hair colors) in various ways, including by hue; this is known as colorimetry. Hue is an interval level variable.
- Cohen's kappa
- Coherence (units of measurement)
- Hume's principle
- Inter-rater reliability
- Logarithmic scale
- Ramsey–Lewis method
- Set theory
- Statistical data type
- Transition (linguistics)
- Kirch, Wilhelm, ed. (2008). "Level of Measurement". Encyclopedia of Public Health. Vol. 2. Springer. pp. 851–852. doi:10.1007/978-1-4020-5614-7_1971. ISBN 978-1-4020-5613-0.
- Stevens, S. S. (7 June 1946). "On the Theory of Scales of Measurement". Science. 103 (2684): 677–680. Bibcode:1946Sci...103..677S. doi:10.1126/science.103.2684.677. PMID 17750512. S2CID 4667599.
- Michell, J. (1986). "Measurement scales and statistics: a clash of paradigms". Psychological Bulletin. 100 (3): 398–407. doi:10.1037/0033-2909.100.3.398.
- Mosteller, Frederick (1977). Data analysis and regression : a second course in statistics. Reading, Mass: Addison-Wesley Pub. Co. ISBN 978-0201048544.
- Chrisman, Nicholas R. (1998). "Rethinking Levels of Measurement for Cartography". Cartography and Geographic Information Science. 25 (4): 231–242. doi:10.1559/152304098782383043. ISSN 1523-0406. – via Taylor & Francis (subscription required)
- Nominal measures are based on sets and depend on categories, a la Aristotle: Chrisman, Nicholas (March 1995). "Beyond Stevens: A revised approach to measurement for geographic information". Retrieved 2014-08-25.
- "Invariably one came up against fundamental physical limits to the accuracy of measurement. ... The art of physical measurement seemed to be a matter of compromise, of choosing between reciprocally related uncertainties. ... Multiplying together the conjugate pairs of uncertainty limits mentioned, however, I found that they formed invariant products of not one but two distinct kinds. ... The first group of limits were calculable a priori from a specification of the instrument. The second group could be calculated only a posteriori from a specification of what was done with the instrument. ... In the first case each unit [of information] would add one additional dimension (conceptual category), whereas in the second each unit would add one additional atomic fact.", – pp. 1–4: MacKay, Donald M. (1969), Information, Mechanism, and Meaning, Cambridge, MA: MIT Press, ISBN 0-262-63-032-X
- Manikandan, S. (2011). "Measures of central tendency: Median and mode". Journal of Pharmacology and Pharmacotherapeutics. 2 (3): 214–5. doi:10.4103/0976-500X.83300. PMC 3157145. PMID 21897729.
- *Lord, Frederic M.; Novick, Melvin R.; Birnbaum, Allan (1968). Statistical Theories of Mental Test Scores. Reading, MA: Addison-Wesley. p. 21. LCCN 68011394.
Although, formally speaking, interval measurement can always be obtained by specification, such specification is theoretically meaningful only if it is implied by the theory and model relevant to the measurement procedure.
- Sheskin, David J. (2007). Handbook of Parametric and Nonparametric Statistical Procedures (Fourth ed.). Boca Raton: Chapman & Hall/CRC. p. 3. ISBN 978-1-58488-814-7.
Although in practice IQ and most other human characteristics measured by psychological tests (such as anxiety, introversion, self esteem, etc.) are treated as interval scales, many researchers would argue that they are more appropriately categorized as ordinal scales. Such arguments would be based on the fact that such measures do not really meet the requirements of an interval scale, because it cannot be demonstrated that equal numerical differences at different points on the scale are comparable.
- Mussen, Paul Henry (1973). Psychology: An Introduction. Lexington (MA): Heath. p. 363. ISBN 978-0-669-61382-7.
The I.Q. is essentially a rank; there are no true "units" of intellectual ability.
- Truch, Steve (1993). The WISC-III Companion: A Guide to Interpretation and Educational Intervention. Austin (TX): Pro-Ed. p. 35. ISBN 978-0-89079-585-9.
An IQ score is not an equal-interval score, as is evident in Table A.4 in the WISC-III manual.
- Bartholomew, David J. (2004). Measuring Intelligence: Facts and Fallacies. Cambridge: Cambridge University Press. p. 50. ISBN 978-0-521-54478-8.
When we come to quantities like IQ or g, as we are presently able to measure them, we shall see later that we have an even lower level of measurement—an ordinal level. This means that the numbers we assign to individuals can only be used to rank them—the number tells us where the individual comes in the rank order and nothing else.
- Eysenck, Hans (1998). Intelligence: A New Look. New Brunswick (NJ): Transaction Publishers. pp. 24–25. ISBN 978-1-56000-360-1.
Ideally, a scale of measurement should have a true zero-point and identical intervals. . . . Scales of hardness lack these advantages, and so does IQ. There is no absolute zero, and a 10-point difference may carry different meanings at different points of the scale.
- Mackintosh, N. J. (1998). IQ and Human Intelligence. Oxford: Oxford University Press. pp. 30–31. ISBN 978-0-19-852367-3.
In the jargon of psychological measurement theory, IQ is an ordinal scale, where we are simply rank-ordering people. . . . It is not even appropriate to claim that the 10-point difference between IQ scores of 110 and 100 is the same as the 10-point difference between IQs of 160 and 150
- Velleman, Paul F.; Wilkinson, Leland (1993). "Nominal, ordinal, interval, and ratio typologies are misleading". The American Statistician. 47 (1): 65–72. doi:10.2307/2684788. JSTOR 2684788.
- Hand, David J. (2017). "Measurement: A Very Short Introduction—Rejoinder to discussion". Measurement: Interdisciplinary Research and Perspectives. 15 (1): 37–50. doi:10.1080/15366367.2017.1360022. hdl:10044/1/50223.
- Nelder, J. A. (1990). The knowledge needed to computerise the analysis and interpretation of statistical information. In Expert systems and artificial intelligence: the need for information about data. Library Association Report, London, March, 23–27.
- van den Berg, G. (1991). Choosing an analysis method. Leiden: DSWO Press
- Wolman, Abel G (2006). "Measurement and meaningfulness in conservation science". Conservation Biology. 20 (6): 1626–1634. doi:10.1111/j.1523-1739.2006.00531.x. PMID 17181798. S2CID 21372776.
- "What is the difference between categorical, ordinal and interval variables?". Institute for Digital Research and Education. University of California, Los Angeles. Archived from the original on 2016-01-25. Retrieved 7 February 2016.
This further reading section may contain inappropriate or excessive suggestions that may not follow Wikipedia's guidelines. Please ensure that only a reasonable number of balanced, topical, reliable, and notable further reading suggestions are given; removing less relevant or redundant publications with the same point of view where appropriate. Consider utilising appropriate texts as inline sources or creating a separate bibliography article. (June 2021)
- Alper, T. M. (1985). "A note on real measurement structures of scale type (m, m + 1)". Journal of Mathematical Psychology. 29: 73–81. doi:10.1016/0022-2496(85)90019-7.
- Alper, T. M. (1987). "A classification of all order-preserving homeomorphism groups of the reals that satisfy finite uniqueness". Journal of Mathematical Psychology. 31 (2): 135–154. doi:10.1016/0022-2496(87)90012-5.
- Briand, L. & El Emam, K. & Morasca, S. (1995). On the Application of Measurement Theory in Software Engineering. Empirical Software Engineering, 1, 61–88. [On line] https://web.archive.org/web/20070926232755/http://www2.umassd.edu/swpi/ISERN/isern-95-04.pdf
- Cliff, N. (1996). Ordinal Methods for Behavioral Data Analysis. Mahwah, NJ: Lawrence Erlbaum. ISBN 0-8058-1333-0
- Cliff, N. & Keats, J. A. (2003). Ordinal Measurement in the Behavioral Sciences. Mahwah, NJ: Erlbaum. ISBN 0-8058-2093-0
- Lord, Frederic M (December 1953). "On the Statistical Treatment of Football Numbers" (PDF). American Psychologist. 8 (12): 750–751. doi:10.1037/h0063675. Archived from the original (PDF) on 20 July 2011. Retrieved 16 September 2010.
- See also reprints in:
- Readings in Statistics, Ch. 3, (Haber, A., Runyon, R. P., and Badia, P.) Reading, Mass: Addison–Wesley, 1970
- Maranell, Gary Michael, ed. (2007). "Chapter 31". Scaling: A Sourcebook for Behavioral Scientists. New Brunswick, New Jersey & London, UK: Aldine Transaction. pp. 402–405. ISBN 978-0-202-36175-8. Retrieved 16 September 2010.
- Hardcastle, G. L. (1995). "S. S. Stevens and the origins of operationism". Philosophy of Science. 62 (3): 404–424. doi:10.1086/289875.
- Lord, F. M., & Novick, M. R. (1968). Statistical theories of mental test scores. Reading, MA: Addison–Wesley.
- Luce, R. D. (1986). "Uniqueness and homogeneity of ordered relational structures". Journal of Mathematical Psychology. 30 (4): 391–415. doi:10.1016/0022-2496(86)90017-9.
- Luce, R. D. (1987). "Measurement structures with Archimedean ordered translation groups". Order. 4 (2): 165–189. doi:10.1007/bf00337695. S2CID 16080432.
- Luce, R. D. (1997). "Quantification and symmetry: commentary on Michell 'Quantitative science and the definition of measurement in psychology'". British Journal of Psychology. 88 (3): 395–398. doi:10.1111/j.2044-8295.1997.tb02645.x.
- Luce, R. D. (2000). Utility of uncertain gains and losses: measurement theoretic and experimental approaches. Mahwah, N.J.: Lawrence Erlbaum.
- Luce, R. D. (2001). "Conditions equivalent to unit representations of ordered relational structures". Journal of Mathematical Psychology. 45 (1): 81–98. doi:10.1006/jmps.1999.1293. PMID 11178923. S2CID 12231599.
- Luce, R. D.; Tukey, J. W. (1964). "Simultaneous conjoint measurement: a new scale type of fundamental measurement". Journal of Mathematical Psychology. 1: 1–27. CiteSeerX 10.1.1.334.5018. doi:10.1016/0022-2496(64)90015-x.
- Michell, J. (1986). "Measurement scales and statistics: a clash of paradigms". Psychological Bulletin. 100 (3): 398–407. doi:10.1037/0033-2909.100.3.398.
- Michell, J. (1997). "Quantitative science and the definition of measurement in psychology". British Journal of Psychology. 88 (3): 355–383. doi:10.1111/j.2044-8295.1997.tb02641.x. S2CID 143169737.
- Michell, J. (1999). Measurement in Psychology – A critical history of a methodological concept. Cambridge: Cambridge University Press.
- Michell, J. (2008). "Is psychometrics pathological science?". Measurement – Interdisciplinary Research & Perspectives. 6 (1–2): 7–24. doi:10.1080/15366360802035489. S2CID 146702066.
- Narens, L. (1981a). "A general theory of ratio scalability with remarks about the measurement-theoretic concept of meaningfulness". Theory and Decision. 13: 1–70. doi:10.1007/bf02342603. S2CID 119401596.
- Narens, L. (1981b). "On the scales of measurement". Journal of Mathematical Psychology. 24 (3): 249–275. doi:10.1016/0022-2496(81)90045-6.
- Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.
- Rozeboom, W. W. (1966). "Scaling theory and the nature of measurement". Synthese. 16 (2): 170–233. doi:10.1007/bf00485356. S2CID 46970420.
- Stevens, S. S. (June 7, 1946). "On the Theory of Scales of Measurement" (PDF). Science. 103 (2684): 677–680. Bibcode:1946Sci...103..677S. doi:10.1126/science.103.2684.677. PMID 17750512. Archived from the original (PDF) on 25 November 2011. Retrieved 16 September 2010.
- Stevens, S. S. (1951). Mathematics, measurement and psychophysics. In S. S. Stevens (Ed.), Handbook of experimental psychology (pp. 1–49). New York: Wiley.
- Stevens, S. S. (1975). Psychophysics. New York: Wiley.
- von Eye, A. (2005). "Review of Cliff and Keats, Ordinal measurement in the behavioral sciences". Applied Psychological Measurement. 29 (5): 401–403. doi:10.1177/0146621605276938.
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